Overbooking is the practice of intentionally
selling more rooms than are available in order to offset the effect of
cancellations and no-shows. Studies estimate that although a hotel is
fully booked, about 5-8% of the rooms are vacant on any given date.
Poor overbooking decisions can prove to be very expensive for the
hotel. In the short run, it is only a loss of room revenue, but over the
long-term, casualties may include decreased customer loyalty, loss of
hotel reputation, etc. American Airlines developed an optimization
model that maximizes net revenues associated with overbooking decisions
for the airline industry.

To illustrate the overbooking model developed by the American
Airlines, let us consider a B757 jet flying from Chicago to Boston. The
aircraft has about 180 seats. Based on the past travel pattern, it is
observed that an average of 5% (or nine passengers) do not turn up at
the time of boarding the flight. If the airlines book all seats for
this leg, it is likely to fly with only 171 occupied seats. However it
does not mean that it never flies with 172 or more (even 180) seats
occupied. There is a lesser chance of the flight flying with 172
passengers, an even lesser chance of it flying with 175 and a miniscule
chance of it flying with all 180
passengers. Therefore, if we book 181 passengers instead of 180, we are
likely to end up with only 173 passengers (and almost always with
lesser than 180 passengers). In an odd event of exactly 181 passengers
reporting, the airline would need to bump one passenger. IATA has
defined rules to compensate bumped passengers. If we can quantify all
costs (including the cost of lost goodwill), the expected revenue would
be the revenue from 181 passengers minus the expected cost of
compensating the one additional passenger at that odd chance. Since the
probability of exactly 181 passengers turning up is so low, the
revenue from that additional passenger generally compensates more than
the expected cost. For this example, the optimal number of passengers
that can be booked would be 186 as illustrated in the figure below.

This model can be directly applied to the hotel industry as well.
The driving force behind the model is the evaluation of the trade off
between additional revenue accrued by selling an already-reserved room
versus the downside from doing so. It has been found that net revenue
increases with overbooking until the point where the downside from
overbooking a room exceeds customer revenues. Beyond that point, the
negative impact of overbooking increases rapidly because fewer and fewer
customers appreciate being turned away.